Finite Group Databasebeta

Subgroups — #19256

Order 256.

345 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	#3077	C₂	1	2	yes	maximal
3	C₂×C₈×D₄	C₂	1	1	yes	maximal, characteristic
4	#1683	C₂	1	4	yes	maximal, complemented
5	#1675	C₂	1	2	yes	maximal, complemented
6	#1669	C₂	1	2	yes	maximal, complemented
7	#1567	C₂	1	1	yes	maximal, characteristic, complemented
8	#1550	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈	—	2	1	no	—
10	#533	C₂²	1	2	yes	—
11	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
12	#500	C₂²	1	1	yes	characteristic, complemented
13	#494	C₂²	1	1	yes	characteristic, complemented
14	#487	C₂²	1	2	yes	complemented
15	#478	—	2	4	no	—
16	#476	C₂²	1	4	yes	—
17	#474	C₂²	1	4	yes	—
18	#467	—	2	1	no	—
19	#467	—	2	1	no	—
20	C₄×Q₁₆	—	2	1	no	—
21	C₄×Q₁₆	—	2	1	no	—
22	C₈×D₄	—	2	1	no	—
23	C₈×D₄	—	2	1	no	—
24	C₈×D₄	C₂²	1	2	yes	—
25	C₈×D₄	C₂²	1	2	yes	—
26	#428	—	2	2	no	—
27	#423	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	#418	—	2	2	no	—
30	C₂×C₂²⋊C₈	—	2	1	no	—
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	#335	C₂²	1	2	yes	complemented
33	#335	C₂²	1	2	yes	complemented
34	#333	C₂²	1	4	yes	complemented
35	#332	C₂²	1	4	yes	complemented
36	C₂²×D₄	D₄	1	1	yes	characteristic
37	C₂³×C₄	—	2	1	no	—
38	C₂×Q₁₆	—	2	1	no	—
39	C₂×Q₁₆	—	2	1	no	—
40	C₂×M₁₆	—	2	2	no	—
41	C₂²×C₈	—	2	1	no	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	—	2	2	no	—
44	C₂²×C₈	—	4	4	no	—
45	C₂²×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₈	D₄	1	1	yes	characteristic
47	C₄⋊Q₈	C₂³	1	2	yes	—
48	C₄⋊Q₈	C₂³	1	2	yes	—
49	#125	—	2	2	no	—
50	C₂²⋊Q₈	—	2	2	no	—
51	C₂²⋊Q₈	—	2	2	no	—
52	C₄×Q₈	—	2	2	no	—
53	C₄×Q₈	—	2	2	no	—
54	C₄×D₄	—	2	1	no	—
55	C₄×D₄	—	2	1	no	—
56	C₄×D₄	D₄	1	2	yes	—
57	C₄×D₄	C₂³	1	2	yes	—
58	#117	—	2	2	no	—
59	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
60	C₂×C₂²⋊C₄	—	2	1	no	—
61	C₂×C₄²	C₂³	1	1	yes	characteristic
62	#107	—	2	1	no	—
63	#107	—	2	1	no	—
64	#107	—	2	4	no	—
65	#106	—	2	4	no	—
66	C₄⋊C₈	—	2	1	no	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	C₂³	1	2	yes	—
69	C₄⋊C₈	C₂³	1	4	yes	—
70	#103	—	2	2	no	—
71	#103	—	2	2	no	—
72	#102	—	2	4	no	—
73	#102	—	2	4	no	—
74	C₂²⋊C₈	—	2	2	no	—
75	C₂²⋊C₈	—	2	2	no	—
76	C₄×C₈	D₄	1	1	yes	characteristic
77	C₄×C₈	D₄	1	1	yes	characteristic
78	C₄×C₈	C₂³	1	1	yes	characteristic
79	C₄×C₈	C₂³	1	1	yes	characteristic
80	C₂⁴	—	2	1	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×D₄	—	2	1	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	Q₁₆	1	4	yes	complemented
86	C₂×D₄	C₂×D₄	1	2	yes	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	4	2	no	—
91	C₂²×C₄	—	4	2	no	—
92	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄	#41	1	1	yes	characteristic
95	Q₁₆	—	4	2	no	—
96	Q₁₆	—	4	2	no	—
97	M₁₆	—	4	4	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	4	4	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	4	4	no	—
105	C₂×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄	1	1	yes	characteristic
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	C₂×D₄	1	2	yes	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₂²⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	4	2	no	—
120	C₄²	—	4	2	no	—
121	C₄²	C₂×D₄	1	1	yes	characteristic
122	C₄²	C₂×D₄	1	1	yes	characteristic
123	C₄²	#41	1	1	yes	characteristic
124	C₄²	C₂⁴	1	1	yes	characteristic
125	C₂³	—	2	2	no	—
126	C₂³	—	2	4	no	—
127	C₂³	—	4	2	no	—
128	C₂³	—	4	2	no	—
129	C₂³	#121	1	1	yes	characteristic
130	Q₈	—	4	2	no	—
131	Q₈	—	4	2	no	—
132	Q₈	—	8	2	no	—
133	D₄	—	2	2	no	—
134	D₄	—	2	4	no	—
135	D₄	C₂×Q₁₆	1	4	yes	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	4	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	#121	1	1	yes	characteristic
147	C₂×C₄	#121	1	1	yes	characteristic
148	C₂×C₄	#121	1	1	yes	characteristic
149	C₂×C₄	C₂×Q₁₆	1	2	yes	—
150	C₂×C₄	#135	1	2	yes	—
151	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄	C₂×#41	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	2	1	no	—
157	C₈	—	4	4	no	—
158	C₈	—	4	2	no	—
159	C₈	AGL(1,ℤ/8)	1	2	yes	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	4	8	no	—
164	C₂²	—	4	2	no	—
165	C₂²	—	4	2	no	—
166	C₂²	#467	1	2	yes	—
167	C₂²	C₂×#121	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	4	2	no	—
172	C₄	—	8	2	no	—
173	C₄	—	8	2	no	—
174	C₄	#467	1	2	yes	—
175	C₄	C₂²×Q₁₆	1	1	yes	characteristic
176	C₄	C₂×#135	1	1	yes	characteristic
177	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₄	#575	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	—	4	4	no	—
181	C₂	#3036	1	1	yes	characteristic
182	C₂	C₂×#467	1	1	yes	characteristic
183	C₂	#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂²×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #3170	C₂	semidirect
4. #1683	C₂	semidirect
5. #1675	C₂	semidirect
6. #1669	C₂	semidirect
7. #1567	C₂	semidirect
8. #1550	C₂	semidirect
12. #500	C₂²	semidirect
13. #494	C₂²	semidirect
14. #487	C₂²	semidirect
32. #335	C₂²	semidirect
33. #335	C₂²	semidirect
34. #333	C₂²	semidirect
35. #332	C₂²	semidirect
85. C₂×D₄	Q₁₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#3170	2
#3077	2
C₂×C₈×D₄	2
#1683	2
#1675	2
#1669	2
#1567	2
#1550	2